
The signal \(x^{\text{PA}}=[x^{\text{PA}}_1,...,x^{\text{PA}}_n,x^{\text{PA}}_{NR}]^{\text{T}}\)  is then sent through a discrete channel with conditional distribution \(P(y|x^{\text{PA}})\) , where \(y\)  is the channel output. Note that elements in \(u_{n:n-K_1}\)  and \(x_{n:n-K_1}\)  with nonpositive indexes are set to zero. Under the assumption of perfect synchronization, the received signal \(y\)  is demodulated via a matched filter and downsampled with a downsampling rate \(R\)  to symbols \(\hat{u}^{\text{sym}} \in \mathbb {C}^{N}\) . Each symbol \(\hat{u}^{\text{sym}}_n\)  is decoded to a probability vector \(\hat{p}_n \in \mathbb {R}^M_+\)  over \(M\)  messages, where \( \sum _{i=1}^M \hat{p}_i=1\) . Finally, the estimated message is obtained by \(\hat{m}_n=\arg \max _m [\hat{p}_n]_{m} \) . Overall, the estimated message sequence \(\hat{m}\)  is obtained from the probability matrix \(\hat{P}=[\hat{p}_1,...,\hat{p}_N] \in \mathbb {R}_+^{M\times N}\) . The demapper can be implemented by an nn as in [1]}, [2]}, [3]}, which can be pretrained to have similar decoding performance as the maximum likelihood demapper.
